Problem: The graph of a sinusoidal function has a maximum point at $(0,8)$ and then has a minimum point at $(5,2)$. Write the formula of the function, where $x$ is entered in radians. $f(x)=$
Answer: The strategy First, let's use the given information to determine the function's amplitude, midline, and period. Then, we should determine whether to use a sine or a cosine function, based on the point where $x=0$. Finally, we should determine the parameters of the function's formula by considering all the above. Determining the amplitude, midline, and period The midline passes exactly between the maximum value $8$ and the minimum value $2$, so the midline equation is $y={5}$. The extremum points are $3$ units above or below the midline, so the amplitude is ${3}$. The minimum point is $5$ units to the right of the maximum point, so the period is $2\cdot 5={10}$. [Why did we multiply by 2?] Determining the type of function to use Since the graph has an extremum point at $x=0$, we should use the cosine function and not the sine function. This means there's no horizontal shift, so the function is of the form $a\cos(bx)+d$. [How do we know that?] Determining the parameters in $a\cos(bx)+d$ Since the extremum point at $x=0$ is a maximum point, we know that $a>0$. [How do we know that?] The amplitude is ${3}$, so $|a|={3}$. Since $a>0$, we can conclude that $a=3$. The midline is $y={5}$, so $d=5$. The period is ${10}$, so $b=\dfrac{2\pi}{{10}}=\dfrac{\pi}{5}$. The answer $f(x)=3\cos\left(\dfrac{\pi}{5}x\right)+5$